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Necklace ring
Ring theory
Ring theory
In mathematics, the necklace ring is a ring introduced by to elucidate the multiplicative properties of necklace polynomials.
Definition
If A is a commutative ring then the necklace ring over A consists of all infinite sequences (a_1, a_2, ...) of elements of A. Addition in the necklace ring is given by pointwise addition of sequences. Multiplication is given by a sort of arithmetic convolution: the product of (a_1, a_2, ...) and (b_1, b_2, ...) has components :\displaystyle c_n=\sum_{[i,j]=n}(i,j)a_ib_j
where [i,j] is the least common multiple of i and j, and (i,j) is their greatest common divisor.
This ring structure is isomorphic to the multiplication of formal power series written in "necklace coordinates": that is, identifying an integer sequence (a_1, a_2, ...) with the power series \textstyle\prod_{n\geq 0} (1{-}t^n)^{-a_n}.
References
This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page.
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